When the Dutch artist Piet Mondrian left Paris in 1938 to escape the threat of a German invasion, he ended up sharing a boarding house in London's Belsize Park – and a new exhibition at the Courtauld gallery shows that the two years he spent in the UK were productive ones, in which he painted some of his purest and most powerful works.

Mondrian || Nicholson: In Parallel traces the relationship between the Dutch modernist master and his friend Ben Nicholson, the British artist who invited him to London. It shows the way Nicholson's work developed in response to Mondrian's, with colourful paintings with wobbly lines giving way to austere white reliefs – one carved out of a table bought from Camden market.

However, letters and other memorabilia displayed at the Courtauld exhibition also reveal that Mondrian was a less forbidding character than his white spaces, precise grids and strict use of primary colours might suggest.

A Snow White and the Seven Dwarves postcard to his brother signed "Sleepy" reveals Mondrian's love of Disney films - he went to see Pinocchio with Nicholson and Barbara Hepworth, Nicholson's second wife, in London.

Mondrian became an avid reader of the London Evening Standard, and – once he had overcome his fear of the steep escalators on the underground – went dancing at London's jazz clubs with the American art collector Peggy Guggenheim, although he was renowned as a terrible mover.

Coincidentally, the Mondrian/Nicholson exhibition arrives in London at the same time as Picasso and Modern British Art at Tate Britain, which charts British responses to another European master and has a section on Nicholson.

"The shows are not some kind of boxing match," Wright said. "They're about how inventive, in relation to European artists, British artists are.

"In this exhibition Nicholson shows himself to be incredibly original and daring. Once he and Mondrian have grown to share this basic aesthetic language, he becomes a really inventive artist.

"We want people to come and understand how the parallel lines that they worked along operated."

Mondrian was particularly nervous about the onset of war, as his work had been included in the Nazis' notorious 1937 show of "degenerate art".

Once settled in London, he wrote in a letter to Nicholson: "Yesterday morning the gouvernment [sic] gave me very kindly a gasmasque [sic] at the town hall. That is not very encouraging."

Mondrian left London in 1940 after his windows had been broken by German bombs in the blitz.

Green said he declined an invitation to move with Nicholson, Hepworth and their children to St Ives, on the grounds that there was "nothing there apart from water". Instead the Dutch artist, an avowed lover of cities, who hated trees and the colour green, moved to New York.

Mondrian's late pictures – such as Broadway Boogie Woogie – will forever associate the artist with that city. However, even after having settled there, he wrote a letter to John Cecil Stephenson, another pioneering British modernist, saying: "I do like New York, but in London I was of course more at home."

Mondrian || Nicholson In Parallel is at the Courtauld Gallery, London, until 20 May